数学代写|密码学代写cryptography theory代考|CS388H

数学代写|密码学代写cryptography theory代考|ELLIPTIC CURVES

Elliptic curves have been studied for a long time by number theorists and a rich and varied theory has been developed. We are interested in elliptic curves because the points on an elliptic curve over a finite field forms a group that is suitable for use in cryptography.

From a mathematical point of view, studying elliptic curves over any field is interesting. From a cryptographic point of view, our groups come from elliptic curves over finite fields, which must therefore be our main interest. To simplify our presentation, we shall restrict ourselves to elliptic curves of a special form defined over prime fields. We note that essentially all of the theory we discuss works equally well for elliptic curves defined over other fields, though sometimes with minor modifications.

Even though we only discuss curves over finite prime fields, it is still convenient to use drawings of curves over the real numbers to illustrate ideas.
We begin by considering the algebraic curve $C$ defined over the field $\mathbb{F}p, p$ a large prime, given by the polynomial equation $$Y^2=X^3+A X+B, \quad A, B \in \mathbb{F}_p .$$ The points on the curve are the points in the affine plane that satisfy the curve equation. However, we cannot restrict the coordinates of the points to be elements of $\mathbb{F}_p$. We fix an algebraic closure $\overline{\mathbb{F}}$ of $\mathbb{F}_p$ and consider the points on the curve to be all the pairs $(x, y) \in \overline{\mathbb{F}}^2$ satisfying the curve equation. A point on a curve with coordinates in $\mathbb{F}_p$ is $\mathbb{F}_p$-rational or just rational. Example 2.24. Consider the curve over $\mathbb{F}{13}$ defined by $Y^2=X^3+X+2$. The points in $\mathbb{F}_p^2$ satisfying the equation are (observe that $11=-2,8=-5$, etc.)
$(1,2),(1,11),(2,5),(2,8),(6,4),(6,9),(7,1),(7,12),(9,5),(9,8),(12,0)$.
A curve is smooth if its partial derivatives never vanish all at the same time for points on the curve.

数学代写|密码学代写cryptography theory代考|Group Operation

We are now ready to turn the set of points on an elliptic curve into a group. We begin by defining a binary operation on the points of the elliptic curve, and then define the actual group operation in terms of the binary operation, as shown in Figure 2.4.

D Definition 2.8. We define two binary operations $*$ and $+$ on an elliptic curve $E$ as: $P * Q$ is the point given by Proposition 2.28, and $P+Q=(P * Q) * \mathcal{O}$.
Example 2.33. Consider the elliptic curve over $\mathbb{F}_{13}$ defined by $Y^2=X^3+X+2$.
Since the unique line through $(1,2)$ and $(7,1)$ intersects the curve in a third point $(9,5)$, we find that $(1,2) *(7,1)=(9,5)$.

The line through $(9,5)$ and $\mathcal{O}$ intersects the curve in $(9,-5)$, so according to the definition $(1,2)+(7,1)=((1,2) *(7,1)) * \mathcal{O}=(9,-5)$.

The unique line through $(6,9)$ that is tangent to the curve intersects the curve in the point $(2,5)$, so $(6,9) *(6,9)-(2,5)$.

The line through $(2,5)$ and $\mathcal{O}$ intersects the curve in $(2,-5)$, so $(6,9)+$ $(6,9)=(2,-5)$.

The unique line through $(2,5)$ and $(2,-5)$ intersects the curve in $\mathcal{O}$. The line through $\mathcal{O}$ and $\mathcal{O}$ is the tangent at $\mathcal{O}$, which intersects only there with multiplicity 3 . In other words, $(2,5)+(2,-5)=\mathcal{O}$.

We shall show that there exists an identity element for $+$, there exists inverses for $+$ and that it is commutative. We shall not show that $+$ is associative. There are many ways to do so, but they are either tedious or advanced.
Exercise 2.63. Let $E$ be an elliptic curve and let $P, Q$ be points on $E$. Show that $P * Q=Q * P$, and consequently that $P+Q=Q+P$.

密码学代考

数学代写|密码学代写cryptography theory代考|ELLIPTIC CURVES

$$Y^2=X^3+A X+B, \quad A, B \in \mathbb{F}_p .$$

$$(1,2),(1,11),(2,5),(2,8),(6,4),(6,9),(7,1),(7,12),(9,5),(9,8),(12,0) \text {. }$$

数学代写|密码学代写cryptography theory代考|Group Operation

$D$ 定义 2.8。我们定义两个二元操作 $*$ 和 $+$ 在椭圆曲线上 $E$ 作为: $P * Q$ 是命题 $2.28$ 给出的点，并且 $P+Q=(P * Q) * \mathcal{O}$.

$$(1,2)+(7,1)=((1,2) *(7,1)) * \mathcal{O}=(9,-5) \text {. }$$

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